Existence of multiple solutions for the laminar flow in a porous channel with suction at both slowly expanding and contracting walls

Abstract

The asymptotic behavior of solutions of a similarity equation for the laminar flow in a porous channel with suction at both expanding and contracting walls has been obtained by using a singular perturbation method. However, in the matching process, this solution neglects exponentially small terms. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find two of the solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions.

Keywords

laminar flow / porous channels / multiple solutions / singular perturbation method

References

[1] Uchida S., Aoki H. Unsteady flows in a semi-infinite contracting or expanding pipe. J. Fluid Mech., 1977, 82(2): 371.

[2] Ohki M. Unsteady flows in a porous, elastic, circular tube: I. The wall contracting or expanding in an axial direction. Bull. JSME, 1980, 23(179): 679.

[3] Goto M., Uchida S. Unsteady flows in a semi-infinite expanding pipe with injection through wall. Trans. Jpn. Soc. Aeronaut. Space Sci., 1990, 33(9): 14.

[4] Bujurke N.M., Pai N.P., Jayaraman G. Computer extended series solution for unsteady flow in a contracting or expanding pipe. IMA J. Appl. Math., 1998, 60(2): 151.

[5] Majdalani J., Zhou C., Dawson C.A. Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. J. Biomech., 2002, 35, 1399.

[6] Majdalani J., Zhou C. Moderate-to-large injection and suction driven channel flows with expanding or contracting walls. Z. Angew. Math. Mech., 2003, 83(3): 181.

[7] Asghar S., Mushtaq M., Hayat T. Flow in a slowly deforming channel with weak permeability: An analytical approach. Nonlinear Anal. Real World Appl., 2010, 11, 555.

[8] Chen Y.L., Zhu K.Q. Couette-Poiseuille flow of Bingham fluids between two porous parallel plates with slip conditions. J. Non-Newton. Fluid Mech., 2008, 153, 1.

[9] Yang S.P., Zhu K.Q. Analytical solutions for squeeze flow of Bingham fluid with Navier slip condition. J. Non-Newton. Fluid Mech., 2006, 138, 173.

[10] Dauenhauer E.C., Majdalani J. Exact self-similarity solution of the Navier-Stokes equations for a porous channel with orthogonally moving walls. Phys. Fluids, 2003, 15(6): 1485.

[11] Berman A.S. Laminar flow in channels with porous walls. J. Appl. Phys., 1953, 24(9): 1232.

[12] Terrill R.M. On some exponentially small terms arising in flow through a porous pipe. Q. J. Mech. Appl. Math., 1973, 26, 347.

[13] Terrill R.M., Thomas P.W. Laminar flow in a uniformly porous pipe. Appl. Sci. Res. A, 1969, 21, 37.

[14] Robinson W.A. The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls. J. Eng. Math., 1976, 10(1): 23.

[15] MacGillivray A.D., Lu C. Asymptotic solution of a laminar flow in a porous channel with large suction: a nonlinear turning point problem. Methods Appl. Anal., 1994, 1(2): 229.

[16] Lu C., MacGillvary A.D., Hastings S.P. Asymptotic behaviour of solutions of a similarity equation for laminar flows in channels with porous walls. IMA J. Appl. Math., 1992, 49, 139.

[17] Zaturska M.B., Drazin P.G., Banks W.H.H. On the flow of a viscous fluid driven along a channel by suction at porous walls. Fluid Dyn. Res., 1988, 4(3): 151.